Instead of applying the decoding framework above to directly reduce
the decoding of a corrupted codeword from \(\mathcal {C}^{\prime }\)  to the unique
decoding problem in \(\mathcal {C}\) , we apply it at each level of a cascade,
reducing the unique decoding problem in \(\mathcal {C}_i\)  to that in
\(\mathcal {C}_{i-1}\) .
If the direct sum at each level of the cascade is an
\((\eta _0, \eta )\) -parity sampler, the list decoding algorithm at radius
\(1/2 - \sqrt{\eta }\)  suffices for unique decoding even if \(\eta \)  is a
(sufficiently small) constant independent of \(\varepsilon \) . This implies that
we can take \(k\)  to be a (suitably large) constant.
This also allows the splittability (and hence \(\lambda \) ) to be
\(2^{-O(k)} = \Omega (1)\) , yielding polynomial rates. We present the
reduction using cascades in
sec:mainresult and the parameter choices in sec:ta-shmaparambasic.
The specific versions of the list decoding results from [1]}
needed here are instantiated in sec:instantiationlistdec.
